Basically if you have Amazon Prime you get Twitch Prime as a new bonus for free.You just have to link your Amazon and Twitch accounts together.

It is with great pride and a tear of joy in our eye that we present to you the newest product collaboration between Twitch and Amazon. Yes, you guessed it, Twitch Prime!!!Since the moment it was announced in 2014 that Amazon would acquire Twitch, you have been wondering how we would work with Amazon to bring value to the Twitch community. Well, we took our cue from you. Right after the acquisition there were Reddit posts and tweets asking when we would offer Twitch Prime:

Here it is: Twitch Prime is a new premium experience on Twitch that is included with Amazon Prime. As a Twitch Prime member, you get free game loot every month, like instant access to the newest Hearthstone hero, Tyrande Whisperwind, or the new indie game, Streamline. You also receive discounts on new-release box games sold by Amazon during the pre-order period and for the first two weeks after launch. And on Twitch, once you link your Amazon Prime account to your Twitch account, you get an ad-free viewing experience, exclusive emotes and chat badge, and one free channel subscription every 30 days. When Twitch Prime members use their free channel subscription every month, the streamer gets paid just like any other subscription, so this introduces a new way to help support the streamers you love, even if you’ve never subscribed before. Or, you can just use it for crashing subscriber-only chat rooms. Your choice, boss!

I have Amazon Prime but I've found no real use for it other than the 2 day shipping. I just signed up for Twitch Prime and see nothing useful there either. I can change the color of the chat and add turbo emoticons now. I've been waiting years for those options! Woohoo!